# Journal of Operator Theory

Volume 75, Issue 1, Winter 2016 pp. 91-118.

Diagonality and Idempotents with applications to problems in
operator theory and frame theory

**Authors**:
Jireh Loreaux (1) and Gary Weiss (2)

**Author institution:** (1) Department of Mathematics, McMicken College of Arts and Sciences,
University of Cincinnati, Cincinnati, Ohio, 45212, U.S.A.

(2) Department of Mathematics, McMicken College of Arts and Sciences,
University of Cincinnati, Cincinnati, Ohio, 45212, U.S.A.

**Summary: **We prove that a nonzero idempotent is zero-diagonal if and only
if it is not a Hilbert-Schmidt perturbation of a projection, along with other useful
equivalences. Zero-diagonal operators are those whose diagonal entries are identically
zero in some basis.
We also prove that any bounded sequence appears as the diagonal of some idempotent operator,
thereby providing a characterization of inner products of dual frame pairs in infinite
dimensions.
Furthermore, we show that any absolutely summable sequence whose sum is a positive
integer appears as the diagonal of a finite rank idempotent.

**DOI: **http://dx.doi.org/10.7900/jot.2014nov05.2054

**Keywords: **idempotents, diagonals, zero-diagonal, Hilbert-Schmidt perturbation, dual frame

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